What's a good neural network for this problem?
A simple feed-forward neural network with at least one hidden layer would suffice in your problem, and can deal with arbitrary non-linear relationships between input and output. If you expect relationships to be highly non-linear then additional layers might be required, but from your description of the problem, I would be surprised if you needed more than few layers, and a relatively small network.
However, I note that:

The input variables are numerous but not that many, maybe 7-10.

This gives you $3^{10} = 59049$ possible inputs. That's not much in terms of amount of data needed for ML statistical models. Assuming that even the best predictions are still probabilistic, then you may only need a million or so examples to create a reasonably accurate lookup table, not needing a neural network at all.
The strength of a neural network is to be able to generalise well from less examples than that. Of course, this is not perfect, but it would be able to do things such as notice if inputs 1,2 and 3 agree then that is always the most likely answer. If that turns out to be true (and not an accident of having low numbers of samples), then the NN could learn that useful pattern using far less data than a table-based approach.

I have read that feedforward nns have limitations regarding control flow and loops, so I'm not sure if that structure will be appropriate.

This is true, but does not impact your situation, because there is no control flow or loops involved. You have described a simple function. Whilst you or I might inspect the data and look backwards and forwards across it before coming to a decision, a neural network approximating a function does not need to do that, and in simple cases there is usually no benefit to doing so - a statistical summary of the correct mapping from input to output is more than sufficient and likely the best that can be done.

I'm not sure if recurrent nn will be appropriate either as I don't care what the previous inputs were.

As all your inputs represent the same kind of thing, you could implement as a RNN with a single input, -1, 0 or +1, always feeding in the predictions by type in the same order. It might resemble how you are thinking about the problem as a human (at least a better analogy than the direct statistical match in a feed-forward network), especially if you implemented a variant of attention. However, I don't think there would be any benefit to that in improved accuracy, and it would be a significant challenge to build that if you are new to AI.